Subgroups — #13092892

Order 1472.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₃	C₂	1	4	yes	maximal, complemented
3	D₄×Dic₂₃	C₂	1	8	yes	maximal, complemented
4	#10625097	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₂₃	C₂²	1	2	yes	—
6	C₂×D₄×C₂₃	C₄	1	8	yes	complemented
7	C₂×D₄×C₂₃	C₂²	1	4	yes	—
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₃	C₂²	1	8	yes	complemented
10	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
11	C₂²⋊Dic₂₃	C₂²	1	8	yes	complemented
12	C₄⋊Dic₂₃	C₂²	1	4	yes	complemented
13	C₄×Dic₂₃	C₂²	1	4	yes	complemented
14	C₂³×C₂₃	—	4	4	no	—
15	C₂³×C₂₃	C₂×C₄	1	8	yes	complemented
16	C₂³×C₂₃	C₂³	1	1	yes	characteristic
17	C₂³×C₂₃	C₂³	1	4	yes	—
18	D₄×C₂₃	C₂×C₄	1	16	yes	complemented
19	C₂×C₄×C₂₃	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₂₃	C₂³	1	2	yes	—
21	C₂×Dic₂₃	—	4	8	no	—
22	C₂×Dic₂₃	C₂³	1	8	yes	—
23	C₂²×C₂₃	—	2	8	no	—
24	C₂²×C₂₃	—	2	2	no	—
25	C₂²×C₂₃	—	4	8	no	—
26	C₂²×C₂₃	C₂²×C₄	1	2	yes	—
27	C₂²×C₂₃	C₂²×C₄	1	8	yes	—
28	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₃	C₂²×C₄	1	4	yes	—
30	Dic₂₃	—	2	8	no	—
31	#518	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	—	2	2	no	—
33	C₂×C₂₃	—	2	8	no	—
34	C₂×C₂₃	C₂³×C₄	1	1	yes	characteristic
35	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
36	C₂²×D₄	D₂₃	1	1	yes	characteristic
37	C₄×D₄	—	23	8	no	—
38	#117	—	23	2	no	—
39	C₂×C₂²⋊C₄	—	23	4	no	—
40	C₂₃	#518	1	1	yes	characteristic, complemented, Sylow
41	C₂⁴	C₂×D₂₃	1	2	yes	—
42	C₂×D₄	Dic₂₃	1	8	yes	complemented
43	C₂×D₄	C₂×D₂₃	1	4	yes	—
44	C₂²×C₄	—	23	8	no	—
45	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
46	C₄⋊C₄	—	23	4	no	—
47	C₂²⋊C₄	—	23	4	no	—
48	C₂²⋊C₄	—	23	8	no	—
49	C₄²	—	23	4	no	—
50	C₂³	—	4	4	no	—
51	C₂³	C₂×Dic₂₃	1	8	yes	complemented
52	C₂³	C₂²×D₂₃	1	1	yes	characteristic
53	C₂³	C₂²×D₂₃	1	4	yes	—
54	D₄	C₂×Dic₂₃	1	16	yes	complemented
55	C₂×C₄	—	23	8	no	—
56	C₂×C₄	—	92	8	no	—
57	C₂×C₄	C₂×Dic₂₃	1	4	yes	complemented
58	C₂×C₄	C₂²×D₂₃	1	2	yes	—
59	C₂²	—	2	8	no	—
60	C₂²	—	2	2	no	—
61	C₂²	—	4	8	no	—
62	C₂²	C₂²×Dic₂₃	1	2	yes	—
63	C₂²	C₂²×Dic₂₃	1	8	yes	—
64	C₂²	C₂³×D₂₃	1	1	yes	characteristic
65	C₄	—	46	8	no	—
66	C₄	C₂²×Dic₂₃	1	4	yes	—
67	C₂	—	2	2	no	—
68	C₂	—	2	8	no	—
69	C₂	#10625155	1	2	yes	—
70	C₂	C₂³×Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator	C₂×C₂₃	C₂³×C₄
Fitting, prime radical	C₂²×D₄×C₂₃	C₂
socle, cosocle	C₂²×C₂₃	C₂⁴
hypercenter	C₂²×D₄	D₂₃
hypocenter	C₂₃	#518
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₂₃	C₂	semidirect
3. D₄×Dic₂₃	C₂	semidirect
4. #10625097	C₂	semidirect
6. C₂×D₄×C₂₃	C₄	semidirect
9. C₂²×Dic₂₃	C₂²	semidirect
11. C₂²⋊Dic₂₃	C₂²	semidirect
12. C₄⋊Dic₂₃	C₂²	semidirect
13. C₄×Dic₂₃	C₂²	semidirect
15. C₂³×C₂₃	C₂×C₄	semidirect
18. D₄×C₂₃	C₂×C₄	semidirect
19. C₂×C₄×C₂₃	C₂×C₄	semidirect
40. C₂₃	#518	semidirect
42. C₂×D₄	Dic₂₃	semidirect
51. C₂³	C₂×Dic₂₃	semidirect
54. D₄	C₂×Dic₂₃	semidirect
57. C₂×C₄	C₂×Dic₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₄×C₂₃	2
C₂×C₂²⋊Dic₂₃	2
D₄×Dic₂₃	2
#10625097	2
#518	23